Differentiate (v^3+5v^2)(2v^7-6v^5) with respect to v.

Answer 1

Method 1

d[(v³+5v²)(2v⁷-6v⁵)]/dv is of the form d (ab) = a'b+ab' where a = v³+5v² and b = 2v⁷-6v⁵

= (3v²+10v)(2v⁷-6v⁵) + (v³+5v²)(14v⁶-30v⁴) = 6v⁹ -18v⁷+20v⁸ -60v⁶+ 14v⁹-30v⁷+70v⁸-150v⁶

= 20v⁹+90v⁸-48v⁷-210v⁶

Method 2 : Expand the term first and you get

= 2v¹⁰ -6v⁸ +10v⁹ -30v⁷ and d(2v¹⁰ -6v⁸ +10v⁹ -30v⁷)/dv = (2)(10)v^10-1 -(6)(8)v^8-1+(10)(9)v^9-1-(30)(7)v^7-1

= 20v⁹ -48v⁷+90v⁸ -210v⁶ rearrange = 20v⁹+90v⁸-48v⁷-210v⁶ Same answer as above